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Every planar graph can be colored with at most four colors such that no two adjacent vertices have the same color.

For a connected planar graph,

A representation of a graph on a surface so that its edges intersect only at their endpoints.

Regions bounded by edges in a planar graph, including the outer (infinite) face.

A planar graph that has been embedded in the plane.

A graph $H$ is a minor of a graph $G$ if $H$ can be obtained from $G$ by deleting edges or vertices and/or contracting edges.

A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of the complete graph $K_5$ or the complete bipartite graph $K_{3,3}$.

A graph that is created by associating a vertex with each face of the original graph and connecting vertices whose corresponding faces in the original graph are adjacent.

A graph that can be embedded in the plane, so that its edges intersect only at their endpoints.